Design‐Sensitivity analysis of Solids Using BEM

Abstract

This paper describes an effective formulation for computing design sensitivities required in the shape optimization of solid objects using the boundary element method (BEM). Implicit differentiation of the discretized boundary integral equations is performed, resulting in a general and efficient analysis technique for design sensitivities of all structural quantities. The numerical integration of kernels is performed, which involves the products of shape functions, fundamental solutions, and their derivatives required for sensitivity calculations. The sensitivities of all components of the boundary stress tensor are obtained without additional numerical integrations. High‐order elements with curved sides are employed for stress and sensitivity analysis. A multizone analysis is implemented and its computational advantages are studied. An approximate method for design sensitivity calculations is also suggested and its performance and computational economy relative to the exact procedure are presented. Comparisons of numerical results for displacement and stress sensitivities, obtained by the present procedure, are made with material derivative results of existing analytical solutions to demonstrate the effectiveness of the present work.